The number of solutions of the equation sin x + sin 3x + sin 5x = 0 in the interval [0, π/2] is:
Correct: C
sin x + sin 3x + sin 5x = 0 => (sin 5x + sin x) + sin 3x = 0 => 2 sin 3x cos 2x + sin 3x = 0 => sin 3x (2 cos 2x + 1) = 0. So either sin 3x = 0 or cos 2x = -1/2. If sin 3x = 0, then 3x = 0, π, 2π, 3π... => x = 0, π/3, 2π/3, π... In the interval [0, π/2], we have x = 0, π/3. If cos 2x = -1/2, then 2x = 2π/3, 4π/3, 8π/3... => x = π/3, 2π/3, 4π/3... In the interval [0, π/2], we have x = π/3. The solutions in [0, π/2] are 0 and π/3. However x=5π/3 is not a valid root. The correct solutions are x = {0,π/3}. Additionally, cos 2x = -1/2. So 2x = 2π/3 => x= π/3; also 2x = 4π/3, so x =2π/3, but it is outside of the interval. So we have the solutions of x =0, x=π/3, so the overall answer is 2